Philo's line, Duplication of cube and new geometric extrema problems

We will discuss the dual of Philo's shortest line segment problem. Philo or Philon of Byzantium found solution of the classical problem of the duplication of cube using this line. This problem asks to find minimal line segment with fixed point and inscribed into a given angle. The dual problem asks to find the optimal line segments passing through two given points, with a common endpoint, and with the other endpoints on a given line. The provided solution uses multivariable calculus and geometry methods. Interesting connections with the angle bisector of the triangle are explored. A generalization of the problem using $L^p$ norm is proposed. Particular case $p=\infty$ is studied. Interesting case $p=2$ is proposed as an open problem and related property of a symedian of a triangle is conjectured. The talk contains many references to historical and new sources.
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